Influence of internal viscosity on the large deformation and buckling of a spherical capsule in a simple shear flow

2011 ◽  
Vol 672 ◽  
pp. 477-486 ◽  
Author(s):  
É. FOESSEL ◽  
J. WALTER ◽  
A.-V. SALSAC ◽  
D. BARTHÈS-BIESEL
Keyword(s):  
Shear Flow ◽  
Simple Shear ◽  
Viscosity Ratio ◽  
Fluid Motion ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Simple Shear Flow ◽  
Flow Strength ◽  
Capsule Deformation ◽  
Internal Viscosity

The motion and deformation of a spherical elastic capsule freely suspended in a simple shear flow is studied numerically, focusing on the effect of the internal-to-external viscosity ratio. The three-dimensional fluid–structure interactions are modelled coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). For low viscosity ratios, the internal viscosity affect the capsule deformation. Conversely, for large viscosity ratios, the slowing effect of the internal motion lowers the overall capsule deformation; the deformation is asymptotically independent of the flow strength and membrane behaviour. An important result is that increasing the internal viscosity leads to membrane compression and possibly buckling. Above a critical value of the viscosity ratio, compression zones are found on the capsule membrane for all flow strengths. This shows that very viscous capsules tend to buckle easily.

Ellipsoidal capsules in simple shear flow: prolate versus oblate initial shapes

2011 ◽  
Vol 676 ◽  
pp. 318-347 ◽  
Author(s):  
J. WALTER ◽  
A.-V. SALSAC ◽  
D. BARTHÈS-BIESEL
Keyword(s):  
Shear Flow ◽  
Simple Shear ◽  
Capillary Number ◽  
Shear Plane ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Simple Shear Flow ◽  
Initial State ◽  
Initial Shape ◽  
Aspect Ratios

The large deformations of an initially-ellipsoidal capsule in a simple shear flow are studied by coupling a boundary integral method for the internal and external flows and a finite-element method for the capsule wall motion. Oblate and prolate spheroids are considered (initial aspect ratios: 0.5 and 2) in the case where the internal and external fluids have the same viscosity and the revolution axis of the initial spheroid lies in the shear plane. The influence of the membrane mechanical properties (mechanical law and ratio of shear to area dilatation moduli) on the capsule behaviour is investigated. Two regimes are found depending on the value of a capillary number comparing viscous and elastic forces. At low capillary numbers, the capsule tumbles, behaving mostly like a solid particle. At higher capillary numbers, the capsule has a fluid-like behaviour and oscillates in the shear flow while its membrane continuously rotates around its deformed shape. During the tumbling-to-swinging transition, the capsule transits through an almost circular profile in the shear plane for which a long axis can no longer be defined. The critical transition capillary number is found to depend mainly on the initial shape of the capsule and on its shear modulus, and weakly on the area dilatation modulus. Qualitatively, oblate and prolate capsules are found to behave similarly, particularly at large capillary numbers when the influence of the initial state fades out. However, the capillary number at which the transition occurs is significantly lower for oblate spheroids.

The dynamics of a vesicle in simple shear flow

2011 ◽  
Vol 674 ◽  
pp. 578-604 ◽  
Author(s):  
HONG ZHAO ◽  
ERIC S. G. SHAQFEH
Keyword(s):  
Shear Flow ◽  
Simple Shear ◽  
Lipid Membrane ◽  
Shear Plane ◽  
Plane Motion ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Lipid Vesicle ◽  
Simple Shear Flow ◽  
Motion Patterns

We have performed direct numerical simulation (DNS) of a lipid vesicle under Stokes flow conditions in simple shear flow. The lipid membrane is modelled as a two-dimensional incompressible fluid with Helfrich surface energy in response to bending deformation. A high-fidelity spectral boundary integral method is used to solve the flow and membrane interaction system; the spectral resolution and convergence of the numerical scheme are demonstrated. The critical viscosity ratios for the transition from tank-treading (TT) to ‘trembling’ (TR, also called VB, i.e. vacillating-breathing, or swinging) and eventually ‘tumbling’ (TU) motions are calculated by linear stability analysis based on this spectral method, and are in good agreement with perturbation theories. The effective shear rheology of a dilute suspension of these vesicles is also calculated over a wide parameter regime. Finally, our DNS reveals a family of time-periodic and off-the-shear-plane motion patterns where the vesicle's configuration follows orbits that resemble but are fundamentally different from the classical Jeffery orbits of rigid particles due to the vesicle's deformability.

Stable equilibrium configurations of an oblate capsule in simple shear flow

2016 ◽  
Vol 791 ◽  
pp. 738-757 ◽  
Author(s):  
C. Dupont ◽  
F. Delahaye ◽  
D. Barthès-Biesel ◽  
A.-V. Salsac
Keyword(s):  
Aspect Ratio ◽  
Shear Flow ◽  
Simple Shear ◽  
Capillary Number ◽  
Stable Equilibrium ◽  
Viscosity Ratio ◽  
Shear Plane ◽  
Initial Orientation ◽  
Simple Shear Flow ◽  
Mechanical Equilibrium

The objective of the paper is to determine the stable mechanical equilibrium states of an oblate capsule subjected to a simple shear flow, by positioning its revolution axis initially off the shear plane. We consider an oblate capsule with a strain-hardening membrane and investigate the influence of the initial orientation, capsule aspect ratio$a/b$, viscosity ratio${\it\lambda}$between the internal and external fluids and the capillary number$Ca$which compares the viscous to the elastic forces. A numerical model coupling the finite element and boundary integral methods is used to solve the three-dimensional fluid–structure interaction problem. For any initial orientation, the capsule converges towards the same mechanical equilibrium state, which is only a function of the capillary number and viscosity ratio. For$a/b=0.5$, only four regimes are stable when${\it\lambda}=1$: tumbling and swinging in the low and medium$Ca$range ($Ca\lesssim 1$), regimes for which the capsule revolution axis is contained within the shear plane; then wobbling during which the capsule experiences precession around the vorticity axis; and finally rolling along the vorticity axis at high capillary numbers. When${\it\lambda}$is increased, the tumbling-to-swinging transition occurs for higher$Ca$; the wobbling regime takes place at lower$Ca$values and within a narrower$Ca$range. For${\it\lambda}\gtrsim 3$, the swinging regime completely disappears, which indicates that the stable equilibrium states are mainly the tumbling and rolling regimes at higher viscosity ratios. We finally show that the$Ca$–${\it\lambda}$phase diagram is qualitatively similar for higher aspect ratio. Only the$Ca$-range over which wobbling is stable increases with$a/b$, restricting the stability ranges of in- and out-of-plane motions, although this phenomenon is mainly visible for viscosity ratios larger than 1.

Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities

1998 ◽  
Vol 361 ◽  
pp. 117-143 ◽  
Author(s):  
S. RAMANUJAN ◽  
C. POZRIKIDIS
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Simple Shear ◽  
Large Deformations ◽  
Viscosity Ratio ◽  
Curvilinear Coordinates ◽  
Triangular Element ◽  
Elastic Membrane ◽  
Simple Shear Flow ◽  
Liquid Drops

The deformation of a liquid capsule enclosed by an elastic membrane in an infinite simple shear flow is studied numerically at vanishing Reynolds numbers using a boundary-element method. The surface of the capsule is discretized into quadratic triangular elements that form an evolving unstructured grid. The elastic membrane tensions are expressed in terms of the surface deformation gradient, which is evaluated from the position of the grid points. Compared to an earlier formulation that uses global curvilinear coordinates, the triangular-element formulation suppresses numerical instabilities due to uneven discretization and thus enables the study of large deformations and the investigation of the effect of fluid viscosities. Computations are performed for capsules with spherical, spheroidal, and discoidal unstressed shapes over an extended range of the dimensionless shear rate and for a broad range of the ratio of the internal to surrounding fluid viscosities. Results for small deformations of spherical capsules are in quantitative agreement with the predictions of perturbation theories. Results for large deformations of spherical capsules and deformations of non-spherical capsules are in qualitative agreement with experimental observations of synthetic capsules and red blood cells. We find that initially spherical capsules deform into steady elongated shapes whose aspect ratios increase with the magnitude of the shear rate. A critical shear rate above which capsules exhibit continuous elongation is not observed for any value of the viscosity ratio. This behaviour contrasts with that of liquid drops with uniform surface tension and with that of axisymmetric capsules subject to a stagnation-point flow. When the shear rate is sufficiently high and the viscosity ratio is sufficiently low, liquid drops exhibit continuous elongation leading to breakup. Axisymmetric capsules deform into thinning needles at sufficiently high rates of elongation, independent of the fluid viscosities. In the case of capsules in shear flow, large elastic tensions develop at large deformations and prevent continued elongation, stressing the importance of the vorticity of the incident flow. The long-time behaviour of deformed capsules depends strongly on the unstressed shape. Oblate capsules exhibit unsteady motions including oscillation about a mean configuration at low viscosity ratios and continuous rotation accompanied by periodic deformation at high viscosity ratios. The viscosity ratio at which the transition from oscillations to tumbling occurs decreases with the sphericity of the unstressed shape. Results on the effective rheological properties of dilute suspensions confirm a non-Newtonian shear-thinning behaviour.

Shear flow over a liquid drop adhering to a solid surface

1996 ◽  
Vol 307 ◽  
pp. 167-190 ◽  
Author(s):  
Xiaofan Li ◽  
C. Pozrikidis
Keyword(s):  
Contact Angle ◽  
Shear Flow ◽  
Contact Line ◽  
Simple Shear ◽  
Liquid Drop ◽  
Contact Angle Hysteresis ◽  
Boundary Integral ◽  
Simple Shear Flow ◽  
Contact Lines ◽  
Transient Deformation

The hydrostatic shape, transient deformation, and asymptotic shape of a small liquid drop with uniform surface tension adhering to a planar wall subject to an overpassing simple shear flow are studied under conditions of Stokes flow. The effects of gravity are considered to be negligible, and the contact line is assumed to have a stationary circular or elliptical shape. In the absence of shear flow, the drop assumes a hydrostatic shape with constant mean curvature. Families of hydrostatic shapes, parameterized by the drop volume and aspect ratio of the contact line, are computed using an iterative finite-difference method. The results illustrate the effect of the shape of the contact line on the distribution of the contact angle around the base, and are discussed with reference to contact-angle hysteresis and stability of stationary shapes. The transient deformation of a drop whose viscosity is equal to that of the ambient fluid, subject to a suddenly applied simple shear flow, is computed for a range of capillary numbers using a boundary-integral method that incorporates global parameterization of the interface and interfacial regriding at large deformations. Critical capillary numbers above which the drop exhibits continued deformation, or the contact angle increases beyond or decreases below the limits tolerated by contact angle hysteresis are established. It is shown that the geometry of the contact line plays an important role in the transient and asymptotic behaviour at long times, quantified in terms of the critical capillary numbers for continued elongation. Drops with elliptical contact lines are likely to dislodge or break off before drops with circular contact lines. The numerical results validate the assumptions of lubrication theory for flat drops, even in cases where the height of the drop is equal to one fifth the radius of the contact line.

Controlling rotation and migration of rings in a simple shear flow through geometric modifications

2018 ◽  
Vol 840 ◽  
pp. 379-407 ◽  
Author(s):  
Neeraj S. Borker ◽  
Abraham D. Stroock ◽  
Donald L. Koch
Keyword(s):  
Shear Flow ◽  
Cross Section ◽  
Simple Shear ◽  
Rigid Bodies ◽  
Boundary Integral ◽  
Simple Shear Flow ◽  
Cross Sectional ◽  
Aspect Ratios ◽  
Gradient Direction ◽  
Continuous Rotation

A ring with a cross-section that has a blunt inner and sharper outer edge can attain an equilibrium orientation in a Newtonian fluid subject to a low Reynolds number simple shear flow. This may be contrasted with the continuous rotation exhibited by most rigid bodies. Such rings align along an orientation when the rotation due to fluid vorticity balances the counter-rotation due to the extensional component of the simple shear flow. While the viscous stress on the particle tries to rotate it, the pressure can generate a counter-vorticity torque that aligns the particle. Using boundary integral computations, we demonstrate ways to effectively control this pressure by altering the geometry of the ring cross-section, thus leading to alignment at moderate particle aspect ratios. Aligning rings that lack fore–aft symmetry can migrate indefinitely along the gradient direction. This differs from the periodic spatial trajectories of fore–aft asymmetric axisymmetric particles that rotate in periodic orbits. The mechanism for migration of aligned rings along the gradient direction is elucidated in this work. The migration speed can be controlled by varying the cross-sectional shape and size of the ring. Our results provide new insights into controlling motion of individual particles and thereby open new pathways towards manipulating macroscopic properties of a suspension.

Numerical simulation of capsule deformation in simple shear flow

2010 ◽  
Vol 39 (2) ◽  
pp. 242-250 ◽  
Author(s):  
Y. Sui ◽  
X.B. Chen ◽  
Y.T. Chew ◽  
P. Roy ◽  
H.T. Low
Keyword(s):  
Numerical Simulation ◽  
Shear Flow ◽  
Simple Shear ◽  
Simple Shear Flow ◽  
Capsule Deformation

Dynamics of capsules enclosing viscoelastic fluid in simple shear flow

2018 ◽  
Vol 840 ◽  
pp. 656-687 ◽  
Author(s):  
Zheng Yuan Luo ◽  
Bo Feng Bai
Keyword(s):  
Shear Flow ◽  
Viscoelastic Fluid ◽  
Simple Shear ◽  
Deformation Behaviour ◽  
Three Dimensional ◽  
High Viscosity ◽  
Deborah Number ◽  
Simple Shear Flow ◽  
Capsule Deformation ◽  
Internal Fluid

Previous studies on capsule dynamics in shear flow have dealt with Newtonian fluids, while the effect of fluid viscoelasticity remains an unresolved fundamental question. In this paper, we report a numerical investigation of the dynamics of capsules enclosing a viscoelastic fluid and which are freely suspended in a Newtonian fluid under simple shear. Systematic simulations are performed at small but non-zero Reynolds numbers (i.e. $Re=0.1$) using a three-dimensional front-tracking finite-difference model, in which the fluid viscoelasticity is introduced via the Oldroyd-B constitutive equation. We demonstrate that the internal fluid viscoelasticity presents significant effects on the deformation behaviour of initially spherical capsules, including transient evolution and equilibrium values of their deformation and orientation. Particularly, the capsule deformation decreases slightly with the Deborah number De increasing from 0 to $O(1)$. In contrast, with De increasing within high levels, i.e. $O(1{-}100)$, the capsule deformation increases continuously and eventually approaches the Newtonian limit having a viscosity the same as the Newtonian part of the viscoelastic capsule. By analysing the viscous stress, pressure and viscoelastic stress acting on the capsule membrane, we reveal that the mechanism underlying the effects of the internal fluid viscoelasticity on the capsule deformation is the alterations in the distribution of the viscoelastic stress at low De and its magnitude at high De, respectively. Furthermore, we find some new features in the dynamics of initially non-spherical capsules induced by the internal fluid viscoelasticity. Particularly, the transition from tumbling to swinging of oblate capsules can be triggered at very high viscosity ratios by increasing De alone. Besides, the critical viscosity ratio for the tumbling-to-swinging transition is remarkably enlarged with De increasing at relatively high levels, i.e. $O(1{-}100)$, while it shows little change at low De, i.e. below $O(1)$.

Flow of a spherical capsule in a pore with circular or square cross-section

2011 ◽  
Vol 705 ◽  
pp. 176-194 ◽  
Author(s):  
X.-Q. Hu ◽  
A.-V. Salsac ◽  
D. Barthès-Biesel
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Cylindrical Tube ◽  
Boundary Integral ◽  
Size Ratio ◽  
Constitutive Law ◽  
Boundary Integral Method ◽  
Flow Strength ◽  
Capsule Deformation

AbstractThe motion and deformation of a spherical elastic capsule freely flowing in a pore of comparable dimension is studied. The thin capsule membrane has a neo-Hookean shear softening constitutive law. The three-dimensional fluid–structure interactions are modelled by coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). In a cylindrical tube with a circular cross-section, the confinement effect of the channel walls leads to compression of the capsule in the hoop direction. The membrane then tends to buckle and to fold as observed experimentally. The capsule deformation is three-dimensional but can be fairly well approximated by an axisymmetric model that ignores the folds. In a microfluidic pore with a square cross-section, the capsule deformation is fully three-dimensional. For the same size ratio and flow rate, a capsule is more deformed in a circular than in a square cross-section pore. We provide new graphs of the deformation parameters and capsule velocity as a function of flow strength and size ratio in a square section pore. We show how these graphs can be used to analyse experimental data on the deformation of artificial capsules in such channels.

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